Effect of Matter Density in the Evolution of the Universe
Kara Farnsworth, Christina King, Ryan Price University of Arizona
The formation, expansion and evolution of the structure of the universe are determined by the ratio of varying densities of the matter components that comprise it. Using the Friedmann Equation to describe the evolution of the universe, we investigate the properties of the each component of the universe to determine the role played by each in the evolution of the universe. Using numerical methods, the Friedmann Equation was solved using various parameters for the densities. From the computations it was found that the age of the universe is mostly dependent upon the dark energy and matter components of the universe while the radiation density has less effect.
Introduction Currently, most scientists believe that the universe is approximately 13.4 billion years old. There are multiple approaches to measuring the age of the universe. One approach is the observation of cosmic microwave background or CMB. By measuring the CMB scientists have studied the radiation of the universe as far back in time as the Big Bang1. The universe is so large that when scientists observe far away materials and radiation, they are in fact looking back in time. This is due to the time taken for light to travel from these places to Earth1. Therefore, by using the amount of radiation of the material to find the distance to these sources, scientists can estimate the age of the universe.
Another method used to find the age of the universe is the rate of expansion of the elements in it. The “discovery that distant galaxies are all receding from us with velocities that are proportional to their distances” shows that all the galaxies came from one single point and expanded at the same time2. By measuring the rate of expansion, assuming that it is constant, and how far apart the objects are today, scientists can estimate the amount of time needed for these astronomical bodies to become as far apart as they are currently3. Using this method the universe is roughly 14 billion years3 old.
The third method is finding specific microwaves from the beginning of the universe and using them to calculate the universe’s age. The “detection of the
1 Barkana, Rennan. "The First Stars in the Universe and Cosmic Reionization." 2 Van Den Bergh, Sidney. "Size and Age of the Universe." 3 Gould, Adam. "IPS Offical Statement on the Ancient Age of the Earth and Universe." Farnsworth, King, Price 2
3 K microwave background by Penzias and Wilson, which represents the red shifted radiation of the primeval fireball” provides evidence of a single starting point and also evidence for the Big Bang, which many scientists believe is how the universe started4. Despite how the universe started, most agree that it has a definite beginning.
There is also other evidence that makes scientists think that there is a single beginning to the universe. The universe could also be “open”, in which there would be no beginning and it would be indefinitely old. However, there is not as much evidence supporting this argument.
Though these methods give an approximate age of the universe, the most exact method is solving Friedmann’s equation. Aleksandr Friedmann was a Russian physicist and mathematician who, after studying Einstein’s Friedmann derived his equation using the Einstein field equations, assuming “a homogeneous and isotropic universe” 5 and wrote the paper On the curvature of Space to Zeitschrift für Physik. In this paper Friedmann proposed that “the radius of curvature of the universe can be either an increasing or a periodic function of time”6 and derived his famous equation.
Assumptions and Equations
While comparing the red-shifts and the distances of different objects in the universe, Edwin Hubble realized there was a relationship between the distance of the object and its velocity away from us. From this information, Hubble concluded the universe is expanding. He expressed this expansion using his “Hubble constant”. v = Hd (1)
Where H is the Hubble constant. With further observation, Hubble realized that the universe is not only expanding, but also accelerating. This means ! that the Hubble constant is not actually constant. If the distance between objects in the universe is represented as R(t), and the distance between objects currently is represented as R0, then the ratio between them,α, is
R(t) !(t) = (2) R0
4 Van Den Bergh, Sidney. "Size and Age of the Universe." 5 Weisstein, Eric W. "Friedmann-Lemaitre Cosmological Model." 6 "Aleksandr Aleksandrovich Friedmann."
Farnsworth, King, Price 3
Where α (0) = 1. Then the Hubble constant can be written in terms of α. 2 1 # d" & H = % ( (3) " $ dt '
By taking the inverse of the present Hubble “constant”, H0, a rough estimate of the age of the universe can be found. !
*1 1 $ 1000m 31536000s 1Mpc ' 10 = t0 " & 71# # # 22 ) =1.34 *10 yrs (4) H0 % 1km 1yr 3*10 m(
Or the universe is approximately 13.4 billion years old. But since the Hubble constant changes with time, this number is only an approximation. The real ! age of the universe is better calculated using the Friedmann equation. The Friedmann equation includes the Hubble constant to give a much more accurate age of the universe.
2 1 # d" & 8)G kc 2 % ( = (*r + *m + *d ) + 2 2 (5) " $ dt ' 3 R0 "
In this equation, G is the gravitational constant = 6.67*10-11Nm2/kg2, ρr is the radiation density of the universe, ρm is the matter density of the universe, ! and ρd is the dark energy density of the universe. The variable k is the curvature parameter of the universe and can be either -1, 0, or 1 for a closed, flat, and open universe respectively. R0 as stated before is the separation distance between objects in the universe presently, and c is the speed of light = 3*108 m/s. The various densities of the universe also vary with time.
" " (t) = r0 (6) r # 4
"m0 "m (t) = 3 (7) ! #
(t) (8) "d = "d 0 ! These densities are usually expressed in terms of the critical density, ρc.
! 3H 2 " = (9) c 8#G
! Farnsworth, King, Price 4
In order to make equation 5 dimensionless, the various densities are better expressed as a ratio of the critical density. The ratios of the density of the universe vs. the critical density are expressed as Ω. # #r (t) r0 "r (t) = = 4 (10) #c #c$
# #m (t) m0 "m (t) = = 3 (11) ! #c #c$
# #d (t) d 0 "d (t) = = (12) ! #c #c
" = "r + "m + "d (13)
! In order for the Friedmann equation to work, there must be a relationship between k (the curvature parameter) and Ω. When Ω < 1, k = -1. When Ω = 1, ! k = 0. When Ω > 1, k =1. Taking these parameters into account, the age of the universe can be found for an open, flat or closed universe. Substituting equations 9, 10, 11 and 12 into equation 5 gives:
2 d# !4 !3 kc = H# (" r# + " m# + " d )! 2 (14) dt R0
This equation was put into a program and be solved for α(t) using the Runge- Kutta method.
Depending on the type of material included in our universe, the universe will have different predicted ages. Research has found that for the universe Hubble’s constant is about 71 km/s/Mpc, r=3x10-5, m=0.27, d=0.73. Using numerical solutions, the age of the universe was investigated for variable values of which were greater than or less than 1. It is important, however, to understand the equations we will be using to find the age of the universe.
The equations used for our research cannot be solved analytically for all solutions. However, using numerical methods, the Friedmann equation, equation 14, can be solved for various density parameters and scenarios. To solve the Friedmann equation numerically, the 4th-order Runge-Kutta method was used. The Runge-Kutta method “propagate(s) a solution over an interval by combining the information from several Euler-style steps (each involving one evaluation of the right-hand f’s), and then using the information obtained to match a Taylor series expansion up to some higher Farnsworth, King, Price 5 order.” 7 The Runge-Kutta method works by using initial values say xn and yn and wanting to find yn+1. The following algorithm6 was used to numerically solve the Friedmann equation:
k1 = h ! f (xn , yn )
' h k1 $ k2 = h ! f % xn + , yn + " & 2 2 #
' h k2 $ k3 = h ! f % xn + , yn + " & 2 2 #
k4 = h ! f (xn + h, yn + k3 ) k k k k y = y + 1 + 2 + 3 + 4 n+1 n 6 3 3 6
Where h is the step size and f(x,y) is the differential equation: the Friedmann equation. Using a given set of initial parameters, the algorithm was able to produce a plot of the value of versus the time t. For each scenario that is numerically solved and plotted, the age of the universe can be inferred from the point where equals zero on the t axis.
Verification of numerical method
The simplest way to verify the numerical method to solve the Friedmann Equation is to look at specific scenarios, formulate their outcomes, and compare them to the numerical results with similar parameters. To do this, one starts with a variation of the Friedmann Equation:
& kc 2 # H 2 = H 2 $( a '4 + ( a '3 + ( ' ! (15) 0 $ r m d 2 2 ! % H 0 a "
Where H is the Hubble parameter described in equation 3. Using this form, there are three specific cases that can be solved analytically. These cases are the matter-dominated universe, the radiation dominated universe and the dark energy dominated universe. For each case, the density parameter was assumed to equal 1 in order to simplify the results.
In the case of a matter-dominated universe, the only dependency is on the -3 of equation 15. By this, Friedmann’s equation becomes:
7 Flannery, Brian P., William H. Press, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes in C: the Art of Scientific Computing. Farnsworth, King, Price 6
2 2 !3 H = H 0 (a )
!3 / 2 H = H 0 (a ) (16)
Substituting equation 3 into equation 16 gives:
( a)da = (H 0 )dt (17)
And upon integrating the terms and using the predefined value of (0) = 1:
2 / 3 & 3H # a(t) = $ 0 (t)! (18) % 2 "
This method of substitution and integration was used with also the dark energy and radiation dominated universes. In the case of dark energy, (t) is given by:
a(t) = exp(H 0t) (19)
And for a radiation dominated universe:
1/ 2 a(t) = (2H 0t) (20)
Equations 18, 19, 20 each provide insight into how each constituent of the universe affects the final plot and acceleration and most importantly, the age of the universe. Each of the equations is plotted in Figure 1 as an example of each variation of the evolution of the universe:
Figure 1 - Plot of Analytical Solutions Farnsworth, King, Price 7
Using these equations, the effect of the numerical method can be checked against these scenarios. To verify the solution, the parameters used for the numerical method is set to the same values as the assumptions used for the analytical solutions. When the numerical method is used to solve for the single dominating factor in the universe, the outcome is closely modeled by the analytical solutions, which verifies the accuracy of the numerical method employed. The plot of the numerical method using the scenarios described above is shown in figure 2:
Figure 2 - Plot of Numerical Solutions
By comparing the two plots it can be seen that both the analytical and numerical plots display the same behavior and thus verifies the legitimacy of the numerical method.
Results
Age of Universe vs. Radiation Density
Ωr t Actual Time (109 years) 14 0.00003 0.998 13.373 13 12
0.1 0.784 10.506 Years) 0.2 0.687 9.206 11
0.3 0.624 8.362 (10^9 10 0.4 0.576 7.718 9 0.5 0.538 7.209 8 Universe 0.6 0.508 6.807 7 of
0.7 0.487 6.526 6 Age 0.8 0.461 6.177 5 0.9 0.442 5.923 0 0.2 0.4 0.6 0.8 1 1.2 1.0 0.432 5.789 Radation Density Table 1 - Radiation Density Time Scale Figure 3 – Radiation Density Timescale Graph Farnsworth, King, Price 8
9 Matter Density Timescale Ωm t Actual Time (10 years) 0.266 0.998 1.337 19 0.1 1.341 1.797 17 0.2 1.095 1.467 Years) 0.3 0.959 1.285 15 0.4 0.867 1.162 (10^9 0.5 0.798 1.069 13 0.6 0.744 0.997 11 0.7 0.701 0.939 Universe
of 0.8 0.671 0.899 9 0.9 0.633 0.848 Age 1.0 0.606 0.812 7 0 0.2 0.4 0.6 0.8 1 1.2 Table 2 – Matter Density Timescale Matter Density Figure 4 – Matter Density Timescale Graph
9 Ωd t Actual Time (10 years) Dark Matter Density Timescale 0.732 0.998 1.337 0.0 1.293 1.733 18 0.1 1.224 1.640 17
0.2 1.170 1.568 Years) 16 0.3 1.127 1.510
0.4 1.090 1.461 (10^9 15
0.5 1.058 1.418 14 0.6 1.031 1.382
0.7 1.006 1.348 Universe 13 of 0.8 0.984 1.319 12
0.9 0.964 1.292 Age 11 1.0 0.945 1.266 0 0.2 0.4 0.6 0.8 1 1.2 Dark Energy Density Table 3 – Dark Matter Timescale Figure 5 – Dark Energy Density Timescale Graph
Figure 3 – Expansion of the Universe for Predicted Values of Density Parameters Farnsworth, King, Price 9
Conclusion
The effects of various densities in the universe were clear from the data produced. From the results, the relationships between each of the components of the universe, its density, and the age of the universe can be determined. The age of the universe has different relationships among each component suggesting that a variation in one parameter would not have the same effect as another parameter.
Many of the results that occurred from solving the Friedmann equation numerically could have been inferred from the analytical solutions. The dependencies upon the value of (t) for each density parameter describe how each component of matter effects the evolution of the universe. The a-4 term describes that the radiation density affects the evolution at small values of t, or the early universe. The radiation at this point causes rapid expansion. As the effect of the radiation decreases on the universe, the influence of the matter with the a-3 dependency increases. At larger values of t, the dark energy becomes dominant causing an accelerating expansion. Because at the current time both the matter density and dark energy densities are playing the largest role in the evolution and acceleration of the universe, they also have much more of a role in age of the universe. As the Friedmann equation is solved backwards in time, the large expansion caused by the two components will also have an important role in how quickly the universe will contract back into a singularity. The results verify this hypothesis as the changes in the matter density and the dark energy density can create a wide range of ages for the universe while the radiation density has a much more limited range in which it can effect the age of the universe (see Tables 1-3 and Figures 3-5).
Farnsworth, King, Price 10
References "Aleksandr Aleksandrovich Friedmann." Dec. 1997. 30 Nov. 2006
Barkana, Rennan. "The First Stars in the Universe and Cosmic Reionization." Science 313 (2006): 931-934. 28 Nov. 2006
Flannery, Brian P., William H. Press, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes in C: the Art of Scientific Computing. 2nd ed. Cambridge: Cambridge UP, 1992.
Gould, Adam. "IPS Offical Statement on the Ancient Age of the Earth and Universe." 12 July 2006. International Planetarium Society. 29 Nov. 2006
Van Den Bergh, Sidney. "Size and Age of the Universe." Science 213 (1981): 825-830. 27 Nov. 2006
Weisstein, Eric W. "Friedmann-Lemaitre Cosmological Model." 2006. 29 Nov. 2006
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